If the current draw is less than full load current (FLA), Yes, you can provided that you don't overload the motor electrically and that the motor can mechanically tolerate the higher-than-normal speed (i.e. its bearings). Bottom line: if power requirements of the driven load don't exceed the rated power of the motor at the higher speed, then there should be no problem in operating the motor at the increased speed.

Short answer: There are losses associated with variable frequency drives of approximately 4%. If the motor is running at full speed and full load via a VFD, the power drawn from the supply will be higher than that of any other form of starter.